Page 15 - DMTH504_DIFFERENTIAL_AND_INTEGRAL_EQUATION

Differential and Integral Equation

Notes 2s n
( 1) s x
J (x) = ...(23)
n (s 1 n ) s 1 2
s 0
,
Thus the general solution of Bessel s equation is
y = A J (x) + B J (x) ...(24)
n n
Where A, B are arbitrary constants.

Example: Proceeding as above shown that for n = 0
2 4 6
x 1 x 1 x
J (x) = 1 ...
0 2 2
2 2 2 3 2
= 1 x 2 x 4 x 6 ... ...(25)
2
2 2 2 4 2 2 2 4 6 2
Prove for integer n
J (x) = J (x) ( 1) n ...(26)
n n
To prove this consider the expression for J (x) i.e.
n
( 1) s x n 2s
J (x) =
n (n s 1) (s 1) 2
s 0
( 1) s x 2s n
Thus J (x) =
n (s 1 n ) (s 1) 2
s 0
n 1 s 2s n s 2s n
( 1) x ( 1) x
= + ...(27)
(s 1 n ) (s 1) 2 (s 1 n ) (s 1) 2
s 0 s n
In the first term we have the argument of
(s + 1 n),

To be negative i.e.
S + 1 n
is ve for s = 0 to n 2 and it is zero for s = n 1. From the properties of gamma functions
(s + 1 n) is for s + 1 n 0 ...(28)
So the first series for J (x) is zero and the expression for J (x) becomes
n n
( 1) s x 2s n
J (x) =
n (s 1 n ) (s 1) 2
s n
Putting S = r + n, we have for
s = n, n + 1,...
r = 0, 1, 2, ...

( 1) r n x 2r n
Thus J (x) =
n (r n 1 n ) (r n 1) 2
r 0

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